The discussion centers on proving whether $\sin(10^{\circ})$ is rational or irrational. It utilizes the relation $\sin \alpha = 3\ \sin \frac{\alpha}{3} - 4\ \sin^{3} \frac{\alpha}{3}$, leading to the cubic equation $8\ x^{3} - 6\ x + 1 =0$. The argument posits that if $x$ were rational, it could be expressed as a fraction of integers, but this leads to a contradiction involving the square root of a non-integer. Consequently, the conclusion drawn is that $\sin(10^{\circ})$ is irrational. The discussion is supported by mathematical reasoning and a shared solution approach.